- Source: SSS*
SSS* is a search algorithm, introduced by George Stockman in 1979, that conducts a state space search traversing a game tree in a best-first fashion similar to that of the A* search algorithm.
SSS* is based on the notion of solution trees. Informally, a solution tree can be formed from any arbitrary game tree by pruning the number of branches at each MAX node to one. Such a tree represents a complete strategy for MAX, since it specifies exactly one MAX action for every possible sequence of moves made by the opponent. Given a game tree, SSS* searches through the space of partial solution trees, gradually analyzing larger and larger subtrees, eventually producing a single solution tree with the same root and Minimax value as the original game tree. SSS* never examines a node that alpha–beta pruning would prune, and may prune some branches that alpha–beta would not. Stockman speculated that SSS* may therefore be a better general algorithm than alpha–beta. However, Igor Roizen and Judea Pearl have shown that the savings in the number of positions that SSS* evaluates relative to alpha/beta is limited and generally not enough to compensate for the increase in other resources (e.g., the storing and sorting of a list of nodes made necessary by the best-first nature of the algorithm). However, Aske Plaat, Jonathan Schaeffer, Wim Pijls and Arie de Bruin have shown that a sequence of null-window alpha–beta calls is equivalent to SSS* (i.e., it expands the same nodes in the same order) when alpha–beta is used with a transposition table, as is the case in all game-playing programs for chess, checkers, etc. Now the storing and sorting of the OPEN list were no longer necessary. This allowed the implementation of (an algorithm equivalent to) SSS* in tournament quality game-playing programs. Experiments showed that it did indeed perform better than Alpha–Beta in practice, but that it did not beat NegaScout.
The reformulation of a best-first algorithm as a sequence of depth-first calls prompted the formulation of a class of null-window alpha–beta algorithms, of which MTD(f) is the best known example.
Algorithm
There is a priority queue OPEN that stores states
(
J
,
s
,
h
)
{\displaystyle (J,s,h)}
or the nodes, where
J
{\displaystyle J}
- node identificator (Dot-decimal notation is used to identify nodes,
ϵ
{\displaystyle \epsilon }
is a root),
s
∈
{
L
,
S
}
{\displaystyle s\in \{L,S\}}
- state of the node
J
{\displaystyle J}
(L - the node is live, which means it's not solved yet and S - the node is solved),
h
∈
(
−
∞
,
∞
)
{\displaystyle h\in (-\infty ,\infty )}
- value of the solved node. Items in OPEN queue are sorted descending by their
h
{\displaystyle h}
value. If more than one node has the same value of
h
{\displaystyle h}
, a node left-most in the tree is chosen.
OPEN := { (e, L, inf) }
while true do // repeat until stopped
pop an element p=(J, s, h) from the head of the OPEN queue
if J = e and s = S then
STOP the algorithm and return h as a result
else
apply Gamma operator for p
Γ
{\displaystyle \Gamma }
operator for
p
=
(
J
,
s
,
h
)
{\displaystyle p=(J,s,h)}
is defined in the following way:
if s = L then
if J is a terminal node then
(1.) add (J, S, min(h, value(J))) to OPEN
else if J is a MIN node then
(2.) add (J.1, L, h) to OPEN
else
(3.) for j=1..number_of_children(J) add (J.j, L, h) to OPEN
else
if J is a MIN node then
(4.) add (parent(J), S, h) to OPEN
remove from OPEN all the states that are associated with the children of parent(J)
else if is_last_child(J) then // if J is the last child of parent(J)
(5.) add (parent(J), S, h) to OPEN
else
(6.) add (parent(J).(k+1), L, h) to OPEN // add state associated with the next child of parent(J) to OPEN
References
External links
Chess Programming Wiki
George Stockman's website